Nonnegative Normal Matrices S. K. Jain and L. E. Snyder Department
of Mathematics
Ohio University Athens, Ohio 45701
Submitted by Richard A. Brualdi
ABSTRACT A structural for which nonnegative
1.
characterization
the transpose coefficients
is given for the class of those nonnegative
is a polynomial and no constant
in the matrix with the polynomial
matrices having
term.
INTRODUCTION
A large number of characterizations of the important class of normal matrices are given by Grone et al. in [2]. One of the characterizations for a real normal matrix is that its transpose is a polynomial in the matrix. A characterization for the class of those stochastic matrices for which the transpose is equal to some power of the matrix is given by Sir&horn in [4]. Our result extends Sinkhorn’s result to any nonnegative A for which AT = (Y, A”’ + . + ak A”‘k, where each cxi > 0 and 0 < rnl < m2 < *** < mk.
2.
DEFINITIONS
AND
NOTATION
A matrix A = (aij> is said to be nonnegative if aij >, 0 for all i,j, and A is said to be positive if aii > 0 for all i, j. A real matrix A is said to be normal if A commutes with AT, the transpose of A. X is said to be a l-inverse of A if AXA = A, and is said to be a {1,2}-inverse of A if in addition XAX = X. A is said to be stochastic if A is nonnegative and the LINEAR
ALGEBRA
AND ITS APPLICATIONS
0 Elsevier Science Publishing Co., Inc., 1993 655 Avenue of the Americas, New York, NY 10010
182: 147-155
(1993) 0024-3795/93/$6.00
147
148
S. K. JAIN AND L. E. SNYDER
sum of each row is one. A = (qj> is called a Z-matrix if aij Q 0 for i # j; A is an M-matrix if A can be expressed in the form A = SI - B, where s > 0, B > 0, and s > p(B), the spectral radius of B.
3.
MAIN
RESULTS
THEOREM 1.
If A is a nonnegative,
nonzero
matrix such that
AT = p(A), where p is a polynomial, p(t)
= altm’
+ CQtm + e-0 +a,YQ,
with all czi > 0, and 0 < ml < m2 < or there is a permutation
*** < mk, then either A is symmetric
matrix P such that
where J is a direct sum of matrices of the following (I)
pxxT,
types:
where x is a positive column vector with xTx = 1,
(II) (
0
0
. . .
0
0
*2x3 T
0
...
0
;
;
0
0
0
...
0
xd- 1xd’
XdXT
0
0
...
0
0
0
x1x2
0 p;
T
where the xi’s are positive unit column vectors, possibly of diferent orders, and P is the unique positive fixed point of the polynomial p. For each summand of type (II) which is a d X d block matrix, for each of the exponents m, in the polynomial p. Proof.
By hypothesis AT =p(A),
d must divide mi + 1
NONNEGATIVE where
NORMAL
149
MATRICES
p is a polynomial,
PW
=
such that oi > 0 and mi > 0 for each i. Then A = c~,[[p(A)]‘~ First we determine
the structure
A = p( AT) = p( p( A)), i.e.,
+ . . . +q[p(A)]? of matrices
(1)
satisfying the polynomial
equa-
tion (1) with positive coefficients. Suppose m, = 1. Then from (1) we obtain (1 -
,;)A
= cyi( crsAn,
+ . . .) + a,[ p( A)]-
+ a.. +q[
p( A)]? (2)
If (pi = 1, then c+ = ~ya = a** = ok = 0, since the left side of (2) is the zero matrix and so k = 1. Hence A is symmetric. If oi # 1, then (pi < 1 since the right side of Equation (2) is nonnegative. Equation (2) thus allows us to express
A as A = A’X,
(3)
where X is a nonnegative matrix which commutes with A. Consequently, A has a nonnegative I-inverse. In case m, > 2, it follows easily from Equation (1) that A can be expressed as in (31, so again A has a nonnegative l-inverse. If X is a nonnegative l-inverse of A, then Y = XAX {l, 2}-inverse of A. Consequently, matrix P such that
by Theorem
where C and D are some nonnegative matrices a direct sum of matrices of the following types:
is a nonnegative
1 of [3] there is a permutation
of appropriate
sizes and ] is
150
S. K. JAIN AND L. E. SNYDER (I)
PxyT, where
x and y are positive vectors with yTx = 1,
(II)
I
PlZW2T
0
0
0
‘..
0
0
“.
0
0
0
P23GY3T
0
0
0
Pdl%iYT
0
0
.
.
.
0
.
.
0
Pd-ldXd-1Y: 0
where xi and yi are positive vectors of the same order with yTxi = 1 and the Pij’s are positive. xi and xj are not necessarily of the same order if i #j. Since AT = p(A), it is also true that MT = PATPT = Pp( A)PT = p(PAPT) = p(M). F rom this it follows that the (2,l) block of M T must be zero matrices, or equivalently, That is, M=PAPT=
block and the (1,s) JO = 0 and Cj = 0.
(‘0 ; i
The matrix J also will satisfy the condition that J’ = p(J), direct sum of matrices implies that each of the summands
and J being a S will likewise
satisfy the condition ST = p(S) for the same polynomial p. Let us consider a summand S of type I, i.e., S = PxyT, with yTx = 1 and x, y being positive vectors. Note that (xyT)j = xyT for all j z 1. Hence pyxT
=
Postmultiply
p( pxyT) = ( a1 pm’ +
p”’
a2
+ ***
+a, p-)
xyT.
(4)
both sides of (4) by x, then (PxTx)y
=
(aJF
+
-0.
+cQ~m~)X.
(5)
It follows that y equals a positive scalar times the positive vector x. Consequently there is no loss in generality in assuming that S has the form S = /3xxT, with x being a positive unit vector. Then (4) becomes
which implies that /3 = p( P> so that P is the unique positive fxed the polynomial p.
point of
NONNEGATIVE
NORMAL
151
MATRICES
Next let us consider a summand when d = 4, then
S of the type II and the powers of S. For
example,
0
S=
0
0
P&Y3T
0
PlZXlY2T
0
0
0
0
0
0
0
&X*YT
P34X3Y4T 0
0
0
0
0
0
0
0
0
0
0
PlZ
0
Yl
&3x1
P23P34QY4T
$2 = P34P41~3Y: 0
,
s3
P41 PlZ
Yz’
0 P23P34P41GYT
=
0
\ and S4
x4
is a block
' I
0
0
PlZ P23P34Xl Y4'
0
0
0
0
0
&P41&X3Yi-
0
O
\
P41&2&3X4Y3T
0
i
'
I
diagonal matrix with the ith block haying the form /.Lx,yT,
where I-L= PiZ Pas P34 P4i. I n general Sd is a block diagonal matrix and the ith block is pxi y,r, where p = &&a ... Pdi. Likewise S qd is a block diagonal matrix for any q, and the ith block is /J~x~yT. From this it follows that for 0 < r < d, Sdq+r = ~4s’. Each of the exponents rni in the polynomial p can be expressed as mi = dqi + ri, where 0 < ri < d. Then
ST = p(S)
Since each (Y~> 0 and only corresponding nonzero blocks Therefore, for any d > 2, of the form mi = dqi + d (6) are equal to
=
c
qyQ’+r’
=
c
(yi
/.,&q’s’:.
the nonzero blocks of Sd-i can match the of ST, it follows that each ri = d - 1. the exponents mi of the polynomial p must be 1. The (1, d) blocks in both sides of Equation
S. K. JAIN AND L. E. SNYDER
152
Multiplying this equation on the right by xd leads to the result that yi is a positive scalar multiple of xi. Similarly it can be shown that each yi is a positive scalar multiple of xi, and so without loss of generality it may be assumed that each xi is a unit vector and that yi = xi for each i. Now let us equate the corresponding
blocks
of the
diagonal
in the
equation SST = STS. For the (1, 1) blocks we have p,“, x1x: = P$ixixT, and for the other blocks we have fi,“, x2 xi = p,“, x2 xi, p,Z, x3 XT = p,“, x3 xi, etc. Hence & = p2a = pa4 = .a* = pdl. Thus the matrix S must have the form
0
x1x2’
0
T
0
...
0
0
.
0
0
0
0
0
0
...
0
0
0
...
0
x2
x3
s=p XdXT
Let C = (l/p>S. Since C”: = CT whenever follows from ST = p(S) that P = p( P>.
Xd-
14
0
I
m, + 1 is a multiple
of d, it n
REMARK 1. If xyT is stochastic, then all components of x are equal. Furthermore, if all corn onents of y are also equal though not necessarily the same as for x, then xy K - Jk, where k is the size of x and Jk is the square matrix of order k with all entries equal to l/k. A normal stochastic matrix A is doubly stochastic (see [4, p. 225]>, i.e., AT is stochastic. Sinkhorn’s structural characterization [4] of those stochastic matrices A for which AT = Ak, follows from Theorem 1 and Remark 1. The conditions in Theorem 1 are also sufficient. THEOREM 2. Zf there is a permutation matrix P such that PAPT is a direct sum of the nonnegative matrices of the types (I) and (II), then there is a monomial p with p( P) = p and AT = p(A).
Proof. Note that if S is a is a periodic matrix, i.e., Cd+l each summand of type (II) summands occur, then A will
summand of type (II), then = C. Let m = Icm(d,, d,, . 1s a dj X di block matrix. be symmetric and the result
S = PC, where C . , d,) - 1, where [If only type (I) is obvious.] Then
NONNEGATIVE S”
=
NORMAL
MATRICES
pmcm = pmCT = pmAIST.
then p(S)
= ST for each summand
153
Hence, if we let p(t) = (l/P”‘-‘It”, S; therefore p(A) = AT,
n
REMARK 2. Note that a positive normal matrix A which is not symmetric cannot satisfy the relation p(A) = AT for any polynomial p with nonnegative coefficients
and no constant
term. For example,
the circulant
matrix
is such a matrix. A natural question arises as to what the structure is for matrices A for which AT = p(A) where p(t) is a polynomial with nonnegative coefficients and p(O) # 0. We have not been able to settle this question, but we give below some observations, contained in Propositions 1 and 2, which may be of interest.
PROPOSITION 1. lfA is a nonnegative
AT= p(A), with cq,, CY~,
where
p(A)
matrix such that
= (~~1 + crlAml + . . . okA”‘k,
, CQ positive, then
(i) A has at most two positive eigenvalues, (ii) A-’ exists and is an M-matrix. Proof.
(i): First we note that if A, is a positive eigenvalue
of A, then
p(h,) is also a positive eigenvalue of A because A and AT have the same eigenvalues. Since the function p is strictly increasing on the interval [O, m), it follows that any positive eigenvalue of A must be a fixed point of p. Also, the graph of p is concave upward on the interval [O, m), and so there can be at most two such fmed points. (ii): From AT = p(A), we have
A = p( AT) = p( p(A))
= a,Z + q[
p( A)]-
+ a** +(Y~[ p( A)lmk,
154
S. K. JAIN AND L. E. SNYDER
so A = p(a,)Z + Aq(A) f or some polynomial cients. Note that p(cz,,> > 0, so
I=
exists and that A-’
N% of [l, p. 1371, A-’
is a Z-matrix [since C$ A) > 01. By
is an M-matrix.
PROPOSITION 2. Zf A is a nonsymmetric AT = p(A), where deg p(t) > 2. Proof.
coeffi-
&[A -A@)] = &[Z - q(A)]A.
This shows that A-’ condition
4 with nonnegative
p(t)
has nonnegative
Assume deg p(t)
= 2. From
R matrix
coefficients
with A 2 0 and and
p(O) # 0,
if
then
A = p( p( A)) we have
O=p(p(A))-A=&,Z-&A+&A2+&A3+&A4p
(7)
where
&,, PI, and p4 are positive and the other P’s are nonnegative. From Proposition 1 we know that A -’ is an M-matrix, so the eigenvalues of A-’ and A must have positive real parts (condition F,, of [l, p. 1501 and that A has at most two real eigenvalues. Hence the minimum polynomial of A must contain a factor t - A, and a factor of the form t2 - at + b, where A,, a, and b are positive. The product of these two factors, say (t - AlXt2 - at + b) = t3 - y2 t2 + y,t - yO, with each -yi > 0 and y0 > 0, must divide the right side of (7). The quotient must be of the form P4(t - h2) with A, > 0, since &, > 0. However, this would then contradict the fact that & > 0. W Hence we must have deg p(t) > 2, completing the proof.
EXAMPLE. The following matrix A satisfies the equation + $A2 and has the eigenvalues
A, = 2 -
fi
AT = +Z + iA
and A, = 2 + fi:
Although it is possible to find larger matrices which satisfy the same polynomial equation in the example above, Proposition 2 shows that any such matrix must be symmetric. The authors have been unable to find a nonsymand metric matrix A and a polynomial p(t) with nonnegative coefficients p(O) # 0 for which
AT = p(A).
NONNEGATIVE
NORMAL
The authors
wish
155
MATRICES
to thank
the referee
for
his helpful
suggestions
and
comments. REFERENCES 1
A.
Berman
and
R. J. Plemmons,
Sciences, Academic, 2
R. Grone, Algebra
3
R.
matrices,
Sinkhorn,
40:225-228
Matrices
in the
Mathematical
E. M. Sa, and H. Wolkowicz,
Trans.
Power
Normal
matrices,
Linear
of nonnegative
group-
(1987).
87:213-225
S. K. Jain, E. K. Kwak, and V. K. Goel, monotone
4
C. R. Johnson,
Appl.
Nonnegative
New York, 1979.
Amer.
symmetric
Math.
Decomposition Sot. 257:371-385
stochastic
matrices,
(1980). Linear
(1981).
Received 24 ]uly 1991; final manuscript
accepted 26 January 1992
Algebra
Appl.